where \({X_j}\)  is any orthonormal basis with respect to the inner product \(\,\cdot \,\,\cdot \,_{\mathfrak {g}}\)  [1]}.
All generators \(A(X)\)  are relatively bounded with respect to the Nelson Laplacian and have relative bound 0. In particular, \(\operatorname{dom}{\Delta }\subset \operatorname{dom}A(X)\) .
The Nelson Laplacian only depends on the inner product but not on the choice of orthonormal basis, see thm:lemmanelson.
For projective unitary representations, the Nelson Laplacian can be defined in a similar way as in eq:nelsonlap. In this case, all its relevant properties from thm:lemmanelson still hold true, see sec:projective.
